Show that $\phi:\mathbb{C}^2\times \mathbb{C}^2\longrightarrow \mathbb{C}$ is a scalar product if and only if $a,d>0$, $c=\bar{b}$ and $ad-|b|^2>0$

Question

I'm stuck with this exercice of scalar products. We define our $\phi:\mathbb{C}^2\times\mathbb{C}^2\longrightarrow\mathbb{C}^2$ $$ \phi(z,w)=(z_1\,\,\,\, z_2)\pmatrix{a&b\\c&d}\pmatrix{\bar{w}_1\\\bar{w}_2} $$ Show that $\phi$ is a scalar product of $\mathbb{C}^2$ if and only if $a,b>0$, $c=\bar{b}$ and $ad-|b|^2>0$.

One implication is not too dificult. I'm stuck showing that if we have the matrix with all the conditions then $\phi(z,z)\geq 0$ and $\phi(z,z)=0\Leftrightarrow z=0$.

All the hints are welcome. Thanks!

Answer 1

Hint $$\phi(z,z)=az_1\bar{z_1}+bz_1\bar{z_2}+cz_2\bar{z_1}+dz_2\bar{z_2} \\ =az_1\bar{z_1}+bz_1\bar{z_2}+\overline{bz_1\bar{z_2}} +dz_2\bar{z_2} \\ $$

Now, since $a>0$ we can put it inside the conjugate

$$\phi(z,z)=a\left( z_1\bar{z_1}+\frac{b}{a}z_1\bar{z_2}+\overline{\frac{b}{a}z_1\bar{z_2}} +\frac{d}{a}z_2\bar{z_2} \right) \\ =a\left( z_1+\overline{\frac{b}{a}}z_2 \right) \overline{\left( z_1+\frac{b}{a}\bar{z_2} \right)}+\mbox{ something }\\ =a\left| z_1+\overline{\frac{b}{a}}z_2 \right|^2+\mbox{ something }$$